Phase retrieval for wavelet transforms

Irène Waldspurger (1, 2)

1 MOKAPLAN - Méthodes numériques pour le problème de Monge-Kantorovich et Applications en sciences sociales
2 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision

This article describes a new algorithm that solves a particular phase retrieval problem, with important applications in audio processing: the reconstruction of a function from its scalogram, that is, from the modulus of its wavelet transform. It is a multiscale iterative algorithm, that reconstructs the signal from low to high frequencies. It relies on a new reformulation of the phase retrieval problem, that involves the holomorphic extension of the wavelet transform. This reformulation allows to propagate phase information from low to high frequencies. Numerical results, on audio and non-audio signals, show that reconstruction is precise and stable to noise. The complexity of the algorithm is linear in the size of the signal, up to logarithmic factors. It can thus be applied to large signals.

Mots clés: multiscale method; iterative algorithms; scalogram; Phase retrieval

Domaines: Optimisation et contrôle [math.OC]

Irène Waldspurger. Phase retrieval for wavelet transforms. IEEE Transactions on Information Theory, 2017. ⟨hal-01645088⟩